On Weak Decay Rates and Uniform Stability of Bounded Linear Operators

TL;DR

This paper investigates conditions under which the spectral radius of a bounded linear operator on a Banach space is less than one, linking decay properties of sequences generated by the operator to spectral stability.

Contribution

It establishes a new criterion for spectral radius less than one based on sequence containment in a principal ideal of $c_0$, generalizing existing theorems.

Findings

Spectral radius $r(T) < 1$ when sequences are in a principal ideal of $c_0$

Generalizations of Weiss and van Neerven's theorems

Results on $C_0$-semigroups related to stability

Abstract

We consider a bounded linear operator $T$ on a complex Banach space $X$ and show that its spectral radius $r(T)$ satisfies $r(T) < 1$ if all sequences $(< x',T^nx>)_{n \in \mathbb{N}_0}$ ($x \in X$, $x' \in X'$) are, up to a certain rearrangement, contained in a principal ideal of the space $c_0$ of sequences which converge to $0$. From this result we obtain generalizations of theorems of G. Weiss and J. van Neerven. We also prove a related result on $C_0$-semigroups.